Inspired by work of Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23 (2019), 249–254] on the minimal excludant or ‘mex’ of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.

A graph G is called an $[s,t]$-graph if any induced subgraph of G of order s has size at least $t.$ We prove that every $2$-connected $[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing results. There are $2$-connected $[4,2]$-graphs which do not satisfy the Chvátal–Erdős condition on Hamiltonicity. We also determine the triangle-free graphs among $[p+2,p]$-graphs for a general $p.$

Thomassen’s chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph G are the length of its longest cycles and the length of its longest chordless cycles, respectively. Harvey [‘A cycle of maximum order in a graph of high minimum degree has a chord’, Electron. J. Combin. 24(4) (2017), Article no. 4.33, 8 pages] proposed the stronger conjecture: every $2$-connected graph G with minimum degree at least $3$ has $c(G)\geq c'(G)+2$. This conjecture implies Thomassen’s chord conjecture. We observe that wheels are the unique Hamiltonian graphs for which the circumference and the induced circumference differ by exactly one. Thus, we need only consider non-Hamiltonian graphs for Harvey’s conjecture. We propose a conjecture involving wheels that is equivalent to Harvey’s conjecture on non-Hamiltonian graphs. A graph is $\ell $-holed if all its holes have length exactly $\ell $. We prove that Harvey’s conjecture and hence also Thomassen’s conjecture hold for $\ell $-holed graphs and graphs with a small induced circumference.

Weighing matrices with entries in the complex cubic and sextic roots of unity are employed to construct Hermitian self-dual codes and Hermitian linear complementary dual codes over the finite field $\mathrm {GF}(4).$ The parameters of these codes are explored for small matrix orders and weights.

Let C and W be two integer sets. If $C+W=\mathbb {Z}$, then we say that C is an additive complement to W. If no proper subset of C is an additive complement to W, then we say that C is a minimal additive complement to W. We study the existence of a minimal additive complement to $W=\{w_i\}_{i=1}^{\infty}$ when W is not eventually periodic and $w_{i+1}-w_{i}\in \{2,3\}$ for all i.

For an integer $k \geq 2$, let $P_{n}^{(k)}$ be the k-generalised Pell sequence, which starts with $0, \ldots ,0,1$ (k terms), and each term thereafter is given by the recurrence $P_{n}^{(k)} = 2 P_{n-1}^{(k)} +P_{n-2}^{(k)} +\cdots +P_{n-k}^{(k)}$. We search for perfect powers, which are sums or differences of two k-generalised Pell numbers.

A class of sequences called L-sequences is introduced, each one being a subsequence of a Collatz sequence. Every ordered pair $(v,w)$ of positive integers determines an odd positive integer P such that there exists an L-sequence of length n for every positive integer n, each term of which is congruent to P modulo $2^{v+w+1}$. The smallest possible initial term of such a sequence is described. If $3^v>2^{v+w}$ the L-sequence is increasing. Otherwise, it is decreasing, except if it is the constant sequence P. A central role is played by Bezout’s identity.

Let G and H be two vertex disjoint graphs. The join $G+H$ is the graph with $V(G+H)=V(G)+V(H)$ and $E(G+H)=E(G)\cup E(H)\cup \{xy\;|\; x\in V(G), y\in V(H)\}$. A (finite) linear forest is a graph consisting of (finite) vertex disjoint paths. We prove that for any finite linear forest F and any nonnull graph H, if $\{F, H\}$-free graphs have a $\chi $-binding function $f(\omega )$, then $\{F, K_n+H\}$-free graphs have a $\chi $-binding function $kf(\omega )$ for some constant k.

Let s be a fixed positive integer constant and let $\varepsilon $ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set ${\mathcal M}\subseteq \mathbb {F}_p^*$ of size $|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $ and integer $H=\lfloor {p^{14/29+\varepsilon }}\rfloor $, any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$

When $s=1$, we show that, for almost all primes p, if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor $ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $, then any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$

Let X be a smooth projective variety defined over a number field K. We give an upper bound for the generalised greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over K. To prove the result, we establish a rather uniform Riemann–Roch-type inequality.

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

Our work owes its origin to a recent note of Ram Murty [‘Irrationality of zeros of the digamma function’, Number Theory in Memory of Eduard Wirsing (eds. H. Maier, R. Steuding and J. Steuding) (Springer, Cham, 2023), 237–243], in which he proves that all the zeros of the digamma function are irrational with at most one possible exception. We extend this investigation to higher-order polygamma functions.

Given a positive integer m, let $\mathbb {Z}_m$ be the set of residue classes mod m. For $A\subseteq \mathbb {Z}_m$ and $n\in \mathbb {Z}_m$, let $\sigma _A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal {H}_m$ be the set of subsets $A\subseteq \mathbb {Z}_m$ such that $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$. Let

$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$

Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove

$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$

and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$.

We demonstrate the existence of K-multimagic squares of order N consisting of $N^2$ distinct integers whenever $N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct kth powers when

$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$

improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].

We develop a generalisation of the square sieve of Heath-Brown and use it to give an alternate proof of one of the large sieve inequalities in our previous paper [‘A large sieve inequality for characters to quadratic moduli’, Preprint, https://web.maths.unsw.edu.au/~ccorrigan/preprint6.pdf].

In this work, we investigate the arithmetic properties of $b_{5^k}(n)$, which counts the partitions of n where no part is divisible by $5^k$. By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.

We prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka, Dharmendra and Kumar [‘Divisibility properties for overcubic partition triples’, Integers 24 (2024), Article no. a80, 9 pages]. We also generalise overcubic partition triples to overcubic partition k-tuples and prove arithmetic properties for these partitions.

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

Let $g(x)=x^3+ax^2+bx+c$ and $f(x)=g(x^3)$ be irreducible polynomials with rational coefficients, and let $ {\mathrm{Gal}}(f)$ be the Galois group of $f(x)$ over $\mathbb {Q}$. We show $ {\mathrm{Gal}}(f)$ is one of 11 possible transitive subgroups of $S_9$, defined up to conjugacy; we use $ {\mathrm{Disc}}(f)$, $ {\mathrm{Disc}}(g)$ and two additional low-degree resolvent polynomials to identify $ {\mathrm{Gal}}(f)$. We further show how our method can be used for determining one-parameter families for a given group. Also included is a related algorithm that, given a field $L/\mathbb {Q}$, determines when L can be defined by an irreducible polynomial of the form $g(x^3)$ and constructs such a polynomial when it exists.

We compute primes $p \equiv 5 \bmod 8$ up to $10^{11}$ for which the Pellian equation $x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes $p\leqslant x$ is well approximated by

$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$

where $\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.